Definition:Field (Abstract Algebra)
This page is about field in abstract algebra. For other uses, see Definition:Field.
Definition
A field is a non-trivial division ring whose ring product is commutative.
Thus, let $\left({F, +, \times}\right)$ be an algebraic structure.
Then $\left({F, +, \times}\right)$ is a field if and only if:
- $(1): \quad$ the algebraic structure $\left({F, +}\right)$ is an abelian group
- $(2): \quad$ the algebraic structure $\left({F^*, \times}\right)$ is an abelian group where $F^* = F \setminus \left\{{0}\right\}$
- $(3): \quad$ the operation $\times$ distributes over $+$.
This definition gives rise to the field axioms, as follows:
Field Axioms
The properties of a field are as follows.
For a given field $\left({F, +, \circ}\right)$, these statements hold true:
\((A0):\) | Closure under addition | \(\displaystyle \forall x, y \in F:\) | \(\displaystyle x + y \in F \) | ||||
\((A1):\) | Associativity of addition | \(\displaystyle \forall x, y, z \in F:\) | \(\displaystyle \left({x + y}\right) + z = x + \left({y + z}\right) \) | ||||
\((A2):\) | Commutativity of addition | \(\displaystyle \forall x, y \in F:\) | \(\displaystyle x + y = y + x \) | ||||
\((A3):\) | Identity element for addition | \(\displaystyle \exists 0_F \in F: \forall x \in F:\) | \(\displaystyle x + 0_F = x = 0_F + x \) | $0_F$ is called the zero | |||
\((A4):\) | Inverse elements for addition | \(\displaystyle \forall x: \exists x' \in F:\) | \(\displaystyle x + x' = 0_F = x' + x \) | $x'$ is called a negative element | |||
\((M0):\) | Closure under product | \(\displaystyle \forall x, y \in F:\) | \(\displaystyle x \circ y \in F \) | ||||
\((M1):\) | Associativity of product | \(\displaystyle \forall x, y, z \in F:\) | \(\displaystyle \left({x \circ y}\right) \circ z = x \circ \left({y \circ z}\right) \) | ||||
\((M2):\) | Commutativity of product | \(\displaystyle \forall x, y \in F:\) | \(\displaystyle x \circ y = y \circ x \) | ||||
\((M3):\) | Identity element for product | \(\displaystyle \exists 1_F \in F, 1_F \ne 0_F: \forall x \in F:\) | \(\displaystyle x \circ 1_F = x = 1_F \circ x \) | $1_F$ is called the unity | |||
\((M4):\) | Inverse elements for product | \(\displaystyle \forall x \in F^*: \exists x^{-1} \in F^*:\) | \(\displaystyle x \circ x^{-1} = 1_F = x^{-1} \circ x \) | ||||
\((D):\) | Product is distributive over addition | \(\displaystyle \forall x, y, z \in F:\) | \(\displaystyle x \circ \left({y + z}\right) = \left({x \circ y}\right) + \left({x \circ z}\right) \) |
These are called the field axioms.
Internationalization
Field is translated:
In Dutch: | lichaam | (literally: body) | ||
In French: | corps | (literally: body) | ||
In German: | Körper | (literally: body) | ||
In Russian: | Поле | (literally: field) | Pronounced: póh-lye | |
In Spanish: | cuerpo | (literally: body) |
Linguistic Note
Note that while in English the word for this entity is field, its name in other European languages translates as body.
The original work was done by Richard Dedekind, who used the word Körper. When translating his work into English, Eliakim Moore mistakenly used the word field. The translator into French did not make the same mistake.
Also see
- Results about fields can be found here.
Sources
- Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): $\S 1.1$
- Murray R. Spiegel: Theory and Problems of Complex Variables (1964)... (previous)... (next): $1$: Complex Numbers: Axiomatic Foundations of the Complex Number System
- Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 23$
- C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 4.14$
- B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970)... (previous)... (next): $\S 1.3$: Some special classes of rings
- A.G. Howson: A Handbook of Terms used in Algebra and Analysis (1972)... (previous)... (next): $\S 6$: Rings and fields
- Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 19$
- Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 55 \ (4)$
- P.M. Cohn: Algebra Volume 1 (2nd ed., 1982)... (previous)... (next): $\S 2.4$: The rational numbers and some finite fields
- James R. Munkres: Topology (2nd ed., 2000)... (previous)... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers